Solutions to Exercises
233
the p 3/2 contour is far from the 0p 3/2 resonance. Consequently, there exists
an optimal contour between these two extremes. It has to be sufficiently far
from basis resonance states for scattering states to have smoothly varying phase
shifts on the Berggren basis contour, and sufficiently close to the real axis for
imaginary parts to remain small as well.
Exercise II.
A. When one deals with bound states, adding more and more Lanczos or JacobiDavidson vectors enlarge the many-body model space in which the Hamiltonian
matrix is diagonalized. Consequently, due to the variational principle, the
eigenstate gains binding energy when the model space is enlarged. However,
there is no such property for unbound states. Indeed, the generalized variational
principle only states that eigenvalues are stationary when the eigenstate is
slightly modified. Energies and widths converge to their exact values by adding
more and more Lanczos or Jacobi-Davidson vectors. However, this convergence
does not necessarily translate into increase of binding energy of the system, as
is the case for bound states.
B. The Berggren basis is complete, so that taking a sufficiently large number
of discretized scattering states insures that completeness will be numerically
precise. Consequently, eigenvalues do not depend on the basis used. As the
eigenenergies of bound states are real, the Hamiltonian matrix represented in
the Berggren basis has real negative eigenvalues, even though it is complex
symmetric. The fact that basis states are complex only implies that subtle
numerical cancellations have to occur in order to obtain real eigenvalues.
C. Narrow resonance states are quasi-stationary, i.e., long lived. Hence, the statistical error on observables A(t) (see Eq. (5.9)), denoted as Δ t (A) in Eq. (5.10), is
small. This explains the small imaginary part of the observables calculated in the
Gamow shell model, which are proportional to statistical errors (see Eq. (5.17)).
On the contrary, broad resonance states have a short lifetime. As a consequence,
the physical observables A(t) associated with broad resonances bear a large
statistical error (see Eqs. (5.10) and (5.17)). This is visible on the complex values
of observable quantities, where both real and imaginary parts are of the same
order of magnitude (see Eqs. (5.16) and (5.17)).
Exercise III.
A. Equations (5.57–5.59) contain only spherical tensor operators. The WignerEckart theorem can then be used to calculate matrix elements. It is thus sufficient
to consider M L = L.
B. One has:
r
L
lab Y LL (Ω lab ) ∝ r
L
lab sin
L (θ lab ) e
iLϕ lab ∝ (x lab + i y lab )
L .
233
the p 3/2 contour is far from the 0p 3/2 resonance. Consequently, there exists
an optimal contour between these two extremes. It has to be sufficiently far
from basis resonance states for scattering states to have smoothly varying phase
shifts on the Berggren basis contour, and sufficiently close to the real axis for
imaginary parts to remain small as well.
Exercise II.
A. When one deals with bound states, adding more and more Lanczos or JacobiDavidson vectors enlarge the many-body model space in which the Hamiltonian
matrix is diagonalized. Consequently, due to the variational principle, the
eigenstate gains binding energy when the model space is enlarged. However,
there is no such property for unbound states. Indeed, the generalized variational
principle only states that eigenvalues are stationary when the eigenstate is
slightly modified. Energies and widths converge to their exact values by adding
more and more Lanczos or Jacobi-Davidson vectors. However, this convergence
does not necessarily translate into increase of binding energy of the system, as
is the case for bound states.
B. The Berggren basis is complete, so that taking a sufficiently large number
of discretized scattering states insures that completeness will be numerically
precise. Consequently, eigenvalues do not depend on the basis used. As the
eigenenergies of bound states are real, the Hamiltonian matrix represented in
the Berggren basis has real negative eigenvalues, even though it is complex
symmetric. The fact that basis states are complex only implies that subtle
numerical cancellations have to occur in order to obtain real eigenvalues.
C. Narrow resonance states are quasi-stationary, i.e., long lived. Hence, the statistical error on observables A(t) (see Eq. (5.9)), denoted as Δ t (A) in Eq. (5.10), is
small. This explains the small imaginary part of the observables calculated in the
Gamow shell model, which are proportional to statistical errors (see Eq. (5.17)).
On the contrary, broad resonance states have a short lifetime. As a consequence,
the physical observables A(t) associated with broad resonances bear a large
statistical error (see Eqs. (5.10) and (5.17)). This is visible on the complex values
of observable quantities, where both real and imaginary parts are of the same
order of magnitude (see Eqs. (5.16) and (5.17)).
Exercise III.
A. Equations (5.57–5.59) contain only spherical tensor operators. The WignerEckart theorem can then be used to calculate matrix elements. It is thus sufficient
to consider M L = L.
B. One has:
r
L
lab Y LL (Ω lab ) ∝ r
L
lab sin
L (θ lab ) e
iLϕ lab ∝ (x lab + i y lab )
L .